The splitting theorem in non-smooth context

TL;DR

This paper proves a splitting theorem for non-smooth metric measure spaces with Ricci curvature bounds, showing such spaces containing a line decompose as a product involving the real line.

Contribution

It extends the classical splitting theorem to the non-smooth setting of infinitesimally Hilbertian CD(0,N) spaces, establishing a fundamental geometric decomposition.

Findings

Spaces with a line split as a product with the real line.

The splitting is stable under measured Gromov-Hausdorff convergence.

The result generalizes the smooth Riemannian case to non-smooth metric measure spaces.

Abstract

We prove that an infinitesimally Hilbertian CD(0,N) space containing a line splits as the product of $R$ and an infinitesimally Hilbertian CD(0,N-1) space. By `infinitesimally Hilbertian' we mean that the Sobolev space $W^{1,2}(X,d,m)$, which in general is a Banach space, is an Hilbert space. When coupled with a curvature-dimension bound, this condition is known to be stable with respect to measured Gromov-Hausdorff convergence.